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A000408
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Numbers that are the sum of three nonzero squares.
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103
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3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 27, 29, 30, 33, 34, 35, 36, 38, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102, 104
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OFFSET
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1,1
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COMMENTS
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a(n) !== 7 (mod 8). - Boris Putievskiy, May 05 2013
A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015
According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 nonzero squares (i.e., is in A000378 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1,2,5,10,13,25,37,58,85,130,?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - Jeffrey Shallit, Jan 15 2017
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 267.
Savin Réalis, Answer to question 25 ("Toute puissance entière de 3 est une somme de trois carrés premiers avec 3"), Mathesis 1 (1881), pp. 87-88. (See also p. 73 where the question is posed.)
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LINKS
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Ray Chandler, Table of n, a(n) for n = 1..10000
Alexander Berkovich and Will Jagy, On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2, arXiv:1101.2951 [math.NT], 2011.
B. Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), J. Int. Seq. 16 (2013) #13.6.4.
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.
S. Mezroui, A. Azizi, and M'hammed Ziane, On a Conjecture of Farhi , J. Int. Seq. 17 (2014) #14.1.8.
Index entries for sequences related to sums of squares
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FORMULA
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a(n) = 6n/5 + O(n/sqrt(log n)). (Can the error term be improved?) - Charles R Greathouse IV, Mar 14 2014
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MAPLE
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N:= 1000: # to get all terms <= N
S:= series((JacobiTheta3(0, q)-1)^3, q, 1001):
select(t -> coeff(S, q, t)>0, [$1..N]); # Robert Israel, Jan 14 2016
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MATHEMATICA
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f[n_] := Flatten[Position[Take[Rest[CoefficientList[Sum[x^(i^2), {i, n}]^3, x]], n^2], _?Positive]]; f[11] (* Ray Chandler, Dec 06 2006 *)
pr[n_] := Select[ PowersRepresentations[n, 3, 2], FreeQ[#, 0] &]; Select[ Range[104], pr[#] != {} &] (* Jean-François Alcover, Apr 04 2013 *)
max = 1000; s = (EllipticTheta[3, 0, q] - 1)^3 + O[q]^(max+1); Select[ Range[max], SeriesCoefficient[s, {q, 0, #}] > 0 &] (* Jean-François Alcover, Feb 01 2016, after Robert Israel *)
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PROG
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(PARI) is(n)=for(x=sqrtint((n-1)\3)+1, sqrtint(n-2), for(y=1, sqrtint(n-x^2-1), if(issquare(n-x^2-y^2), return(1)))); 0 \\ Charles R Greathouse IV, Apr 04 2013
(PARI) is(n)= my(a, b) ; a=1 ; while(a^2+1<n, b=1 ; while(b<=a && a^2+b^2<n, if(issquare(n-a^2-b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0) ;
for(n=3, 1e3, if(is(n), print1(n, ", "))); \\ Altug Alkan, Jan 18 2016
(Haskell)
a000408 n = a000408_list !! (n-1)
a000408_list = filter ((> 0) . a025427) [1..]
-- Reinhard Zumkeller, Feb 26 2015
(Python)
def aupto(lim):
squares = [k*k for k in range(1, int(lim**.5)+2) if k*k <= lim]
sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
sum3sqs = set(a+b for a in sum2sqs for b in squares)
return sorted(set(range(lim+1)) & sum3sqs)
print(aupto(104)) # Michael S. Branicky, Mar 06 2021
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CROSSREFS
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Cf. A000378, A000404, A000414, A003072, A004214, A024795, A024796, A025321, A025427.
Sequence in context: A065940 A358350 A024795 * A025321 A153238 A343112
Adjacent sequences: A000405 A000406 A000407 * A000409 A000410 A000411
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane and J. H. Conway
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STATUS
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approved
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